Gravitation Potential Energy — Question 1

Question 1

Equal altitude steps require unequal amounts of lifting work

A small mass $m$ is raised quasistatically outside a nonrotating spherical body of mass $M$ and radius $R$. Use $U(r)=-GMm/r$, zero at infinity.

1. Find external work from $R$ to $2R$ and from $2R$ to $3R$.

2. Compare their sum with the constant-surface-gravity estimate for the same total altitude gain.

Question 1 - Solution

Equal altitude steps require unequal amounts of lifting work

(a) Derivation and calculation

Quasistatic motion has $\Delta K=0$, so external work equals $\Delta U$: $$W_{a\to b}=U(r_b)-U(r_a)=GMm\left(\frac1{r_a}-\frac1{r_b}\right).$$ Thus $$\boxed{W_1=GMm(1/R-1/2R)=\frac{GMm}{2R}},$$ $$\boxed{W_2=GMm(1/2R-1/3R)=\frac{GMm}{6R}}=W_1/3.$$

(b) Result and interpretation

The exact total and constant-gravity estimate are $$\boxed{W_{\mathrm{exact}}=W_1+W_2=\frac{2GMm}{3R}},$$ $$W_{g_s}=mg_s(2R)=\frac{2GMm}{R},\qquad \boxed{W_{g_s}/W_{\mathrm{exact}}=3}.$$ Gravity weakens during the lift; using its surface value over the entire distance overestimates the required work.

Physical check

Both segment works are positive, and their sum equals the direct potential change from $R$ to $3R$.